Matrix elements of the interacting real scalar field $\varphi(x)$ differ from the matrix elements of the in-(scalar) fields (which follow the free Klein-Gordon equation and are the asymptotic fields at infinity) by (Bjorken & Drell shortly B & D, the tags are also B&D ):
$$\langle 0| \varphi(x)|p\rangle=\sqrt{Z}\langle 0| \varphi_{in}(x)|p\rangle .\tag{B&D 16.37}$$
The physical interpretation of this is that an interacting field $\varphi(x)$ if acting on the vacuum can not only generate 1-particle states, but also various multi-particle states. Therefore $\sqrt{Z}$ corresponds to the portion of 1-particle states that is generated by the interacting $\varphi$. Already this interpretation suggest that $\sqrt{Z}$ should have a value of smaller than 1.
In particular according to Bjorken & Drell (B&D) chapter 16.4 the vacuum expectation value of the commutator for the interacting real scalar quantum field theory is defined as:
$$i\Delta'(x,x')=i\Delta'(x-x')=\langle 0| [\varphi(x),\varphi(x')]|0\rangle. \tag{B&D 16.23 + 16.25}$$
According to the Källen-Lehmann spectral representation it can be written in the following way:
$$i\Delta'(x-x') = Zi\Delta(x-x',m) + i\int_{m_1^2}^\infty \ d\sigma^2 \rho(\sigma^2)\Delta(x-x';\sigma) \tag{B&D 16.40}$$
where the threshold (value) $m_1^2$ is now the mass square of the lightest state in the continuum above the discrete 1-particle state which contributes to $\rho(\sigma^2)$. $\rho(\sigma^2)$ is a positive spectral density function which can be computed as a sum of particle states $\lambda$:
$$\rho(\sigma^2) = \sum\limits_\lambda (2\pi)\delta(\sigma^2-m^2_\lambda)|\langle 0 |\varphi(0)|\lambda\rangle |^2.$$
BTW, a similar formula, however for the Feynman-propagator can be found in Peskin & Schroeder in chapter 7.1 formula (7.9).
In order to find a relationship for $Z$, the renormalization "constant", Bjorken&Drell take the time derivative $t$ and the limit $t'\rightarrow t$:
$$\lim_{t'\rightarrow t}\left(\frac{\partial}{\partial t} i\Delta'(x-x') \right)=\langle 0|[\dot{\varphi}(\mathbf{x},t),\varphi(\mathbf{x}',t)]|0\rangle = -i \delta^{(3)}(\mathbf{x}-\mathbf{x}') \tag{B&D 16.41}$$
A very similar relationship is also true for the vacuum expectation value of the commutator of the non-interacting scalar field:
$$ \lim_{t'\rightarrow t}\left(\frac{\partial}{\partial t} i\Delta(x-x',m) \right) = -i \delta^{(3)}(\mathbf{x}-\mathbf{x}'). \tag{B&D 12.42}$$
If both preceding results are plugged in the time derivative of the Källen-Lehmann spectral representation and the delta-function is cancelled out we get:
$$1 = Z +\int_{m_1^2}^\infty \rho(\sigma^2) d\sigma^2 \tag{B&D 16.42}$$
from which we conclude that the renormalization "constant" Z has positive values below 1:
$$ 0\leq Z < 1 \tag{B&D 16.43}$$
because as we can very well assume that the spectral density function is positive.
However, in the renormalization procedure it turns out the $Z$ can adopt arbitrary high values much larger than 1 if not infinite if choose to shift the cutoff $\Lambda\rightarrow\infty$. How can with this contradiction be dealt?
